| L(s) = 1 | − 2.04·2-s − 9·3-s − 27.8·4-s + 42.3·5-s + 18.3·6-s + 191.·7-s + 122.·8-s + 81·9-s − 86.4·10-s − 655.·11-s + 250.·12-s − 932.·13-s − 391.·14-s − 381.·15-s + 641.·16-s − 344.·17-s − 165.·18-s − 548.·19-s − 1.17e3·20-s − 1.72e3·21-s + 1.33e3·22-s − 529·23-s − 1.09e3·24-s − 1.33e3·25-s + 1.90e3·26-s − 729·27-s − 5.33e3·28-s + ⋯ |
| L(s) = 1 | − 0.360·2-s − 0.577·3-s − 0.869·4-s + 0.757·5-s + 0.208·6-s + 1.47·7-s + 0.674·8-s + 0.333·9-s − 0.273·10-s − 1.63·11-s + 0.502·12-s − 1.53·13-s − 0.533·14-s − 0.437·15-s + 0.626·16-s − 0.288·17-s − 0.120·18-s − 0.348·19-s − 0.659·20-s − 0.853·21-s + 0.589·22-s − 0.208·23-s − 0.389·24-s − 0.425·25-s + 0.552·26-s − 0.192·27-s − 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 2.04T + 32T^{2} \) |
| 5 | \( 1 - 42.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 191.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 655.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 932.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 344.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 548.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.43e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.28e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.56e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.51e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.21e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.80e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.83e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.37e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.22e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.27390064902417706729492418350, −12.10157549381494325397413681497, −10.62774457941911726983967941440, −9.973480539145993025147828164896, −8.486587045428720091537179118244, −7.46318103600097050716593551671, −5.33551902946557951944024379568, −4.79951422986388450327561804934, −1.96789499615044439245434461760, 0,
1.96789499615044439245434461760, 4.79951422986388450327561804934, 5.33551902946557951944024379568, 7.46318103600097050716593551671, 8.486587045428720091537179118244, 9.973480539145993025147828164896, 10.62774457941911726983967941440, 12.10157549381494325397413681497, 13.27390064902417706729492418350
